Orthogonal Geodesic and Minimal Distributions

Abstract

Abstract. Let 8 be a smooth distribution on a Riemannian manifold M with $ the orthogonal distribution. We say that 5 is geodesic provided g is integrable with leaves which are totally geodesic submanifolds of M. The notion of minimality of a submanifold of M may be defined in terms of a criterion involving any orthonormal frame field tangent to the given submanifold. If this criterion is satisfied by any orthonormal frame field tangent to ip then we say ip is minimal. Suppose that 8 and § are orthogonal geodesic and minimal distributions on a submanifold of Euclidean space. Then each leaf of 3 is also a submanifold of Euclidean space with mean curvature normal vector field r¡. We show that the integral of | ij |2 over M is bounded below by an intrinsic constant and give necessary and sufficient conditions for equality to hold.We study the relationships between the geometry of M and the integrability of §. For example, if 3 and § are orthogonal geodesic and minimal distributions on a space of nonnegative sectional curvature then § is integrable iff 3 and § are parallel distributions. Similarly if M" has constant negative sectional curvature and dim § = 2 < n then § is not integrable. If 3 is geodesic and § is integrable then we characterize the local structure of the Riemannian metric in the case that the leaves of § are flat submanifolds of A/, with parallel second fundamental form.

On the geometry of harmonic morphisms

“…From Proposition 26, A is constant and 2tf is integrable, and we have assumed that T = 0, that is, U is totally geodesic. It then follows from theorem 1-6 of [14] that 3ff is totally geodesic, and then by theorem 3.2 of [7], that U and JV are parallel. Let H o c U m be the n-dimensional subspace of fR m parallel to the leaves of Jtif.…”